AS DEPENDING ON THE SYMBOLIC EQUATION 6 n = 1. 
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125] 
Again, in the theory of matrices, if I denote the operation of inversion, and tr 
that of transposition, (I do not stop to explain the terms as the example may be 
passed over), we may write , 
a = I, (3 = tr, 7 = /. tr = tr. 
I proceed to the case of a group of six symbols, 
l, a , A % 8, e, 
which may be considered as representing a system of roots of the symbolic equation 
6 s = 1. 
It is in the first place to be shown that there is at least one root which is a 
prime root of 0 3 = l, or (to use a simpler expression) a root having the index 3. It 
is clear that if there were a prime root, or root having the index 6, the square of 
this root would have the index 3, it is therefore only necessary to show that it is 
impossible that all the roots should have the index 2. This may be done by means 
of a theorem which I shall for the present assume, viz. that if among the roots of 
the symbolic equation 6 n = l, there are contained a system of roots of the symbolic 
equation 6 p — 1 (or, in other words, if among the symbols forming a group of the order 
there are contained symbols forming a group of the order p), then p is a submultiple 
of n. In the particular case in question, a group of the order 4 cannot form part 
of the group of the order 6. Suppose, then, that 7, 8 are two roots of 0 6 = 1, having 
each of them the index 2; then if 78 had also the index 2, we should have 78 = 87; 
and l, 7, 8, 87, which is part of the group of the order 6, would be a group of 
the order 4. It is easy to see that 78 must have the index 3, and that the group 
is, in fact, l, 78, 87, 7, 8, 787, which is, in fact, one of the groups to be presently 
obtained; I prefer commencing with the assumption of a root having the index 3. 
Suppose that a is such a root, the group must clearly be of the form 
l, a, a 2 , 7, ary, a 2 y, (a 3 =l); 
and multiplying the entire group by 7 as nearer factor, it becomes 7, ay, dry, y' 2 , 
ay 2 , a 2 y 2 ; we must therefore have 7 s = 1, a, or a 2 . But the supposition y 2 = a 2 gives 
7 4 = a i = a, and the group is in this case l, 7, y 2 , 7 3 , y i , 7® (70=1); and the suppo 
sition 7 2 = a gives also this same group. It only remains, therefore, to assume 7 s = l; 
then we must have either 7a — ay or else ya — a 2 y. The former assumption leads to 
the group 
1, a, a 2 , 7, ay, a 2 y, (a 3 = l, 7 2 = l, ya = ay), 
which is, in fact, analogous to the system of roots of the ordinary equation ¿c 6 — 1 = 0; 
and by putting ay = \, might be exhibited in the form 1, A, A 2 , A 3 , A 4 , A 5 , (A 6 =l), 
under which this system has previously been considered. The latter assumption leads 
to the group 
l, a, a 2 , 7, ay, a 2 y, (a s = l, y 2 = 1, ya = d 2 y), 
and we have thus two, and only two, essentially distinct forms of a group of six. 
If we represent the first of these two forms, viz. the group 
1, a, a 2 , 7, ay, a 2 y, (a 3 = l, 7 2 = 1, ya = ay)